Presentation of special series with computed recurrently coefficients of solutions of nonlinear evolution equations

One of the analytical methods of presenting solutions of nonlinear partial differential equations is the method of special series in powers of specially selected functions called basis functions. The coefficients of such series are found successively as solutions of linear differential equations. To find recurrence, the coefficient is achieved by the choice of basis functions, which may also contain arbitrary functions. By using such functional arbitrariness, it allows in some cases to prove the global convergence of the corresponding constructed series, as well as the solvability of the boundary value problem.